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.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "examples/mdo/plot_sellar.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        Click :ref:`here <sphx_glr_download_examples_mdo_plot_sellar.py>`
        to download the full example code

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_examples_mdo_plot_sellar.py:


A from scratch example on the Sellar problem
============================================
.. _sellar_from_scratch:

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Introduction
------------
In this example,
we will create an MDO scenario based on the Sellar's problem from scratch.
Contrary to the :ref:`GEMSEO in ten minutes tutorial <gemseo_10min>`,
all the disciplines will be implemented from scratch
by sub-classing the :class:`.MDODiscipline` class
for each discipline of the Sellar problem.

The Sellar problem
------------------
We will consider in this example the Sellar problem:

.. include:: /tutorials/_description/sellar_problem_definition.inc

Imports
-------
All the imports needed for the tutorials are performed here.

.. GENERATED FROM PYTHON SOURCE LINES 44-59

.. code-block:: default

    from __future__ import annotations

    from math import exp

    from gemseo.algos.design_space import DesignSpace
    from gemseo.api import configure_logger
    from gemseo.api import create_scenario
    from gemseo.core.discipline import MDODiscipline
    from matplotlib import pyplot as plt
    from numpy import array
    from numpy import ones

    configure_logger()






.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    <RootLogger root (INFO)>



.. GENERATED FROM PYTHON SOURCE LINES 60-86

Create the disciplinary classes
-------------------------------
In this section,
we define the Sellar disciplines by sub-classing the :class:`.MDODiscipline` class.
For each class,
the constructor and the _run method are overriden:

- In the constructor,
  the input and output grammar are created.
  They define which inputs and outputs variables are allowed
  at the discipline execution.
  The default inputs are also defined,
  in case of the user does not provide them at the discipline execution.
- In the _run method is implemented the concrete computation of the discipline.
  The inputs data are fetch
  by using the :meth:`.MDODiscipline.get_inputs_by_name` method.
  The returned NumPy arrays can then be used to compute the output values.
  They can then be stored in the :attr:`!MDODiscipline.local_data` dictionary.
  If the discipline execution is successful.

Note that we do not define the Jacobians in the disciplines.
In this example,
we will approximate the derivatives using the finite differences method.

Create the SellarSystem class
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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.. code-block:: default



    class SellarSystem(MDODiscipline):
        def __init__(self):
            super().__init__()
            # Initialize the grammars to define inputs and outputs
            self.input_grammar.update(["x", "z", "y_1", "y_2"])
            self.output_grammar.update(["obj", "c_1", "c_2"])
            # Default inputs define what data to use when the inputs are not
            # provided to the execute method
            self.default_inputs = {
                "x": ones(1),
                "z": array([4.0, 3.0]),
                "y_1": ones(1),
                "y_2": ones(1),
            }

        def _run(self):
            # The run method defines what happens at execution
            # ie how outputs are computed from inputs
            x, z, y_1, y_2 = self.get_inputs_by_name(["x", "z", "y_1", "y_2"])
            # The ouputs are stored here
            self.local_data["obj"] = array([x[0] ** 2 + z[1] + y_1[0] ** 2 + exp(-y_2[0])])
            self.local_data["c_1"] = array([3.16 - y_1[0] ** 2])
            self.local_data["c_2"] = array([y_2[0] - 24.0])









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Create the Sellar1 class
^^^^^^^^^^^^^^^^^^^^^^^^

.. GENERATED FROM PYTHON SOURCE LINES 116-137

.. code-block:: default



    class Sellar1(MDODiscipline):
        def __init__(self):
            super().__init__()
            self.input_grammar.update(["x", "z", "y_2"])
            self.output_grammar.update(["y_1"])
            self.default_inputs = {
                "x": ones(1),
                "z": array([4.0, 3.0]),
                "y_1": ones(1),
                "y_2": ones(1),
            }

        def _run(self):
            x, z, y_2 = self.get_inputs_by_name(["x", "z", "y_2"])
            self.local_data["y_1"] = array(
                [(z[0] ** 2 + z[1] + x[0] - 0.2 * y_2[0]) ** 0.5]
            )









.. GENERATED FROM PYTHON SOURCE LINES 138-140

Create the Sellar2 class
^^^^^^^^^^^^^^^^^^^^^^^^

.. GENERATED FROM PYTHON SOURCE LINES 140-159

.. code-block:: default



    class Sellar2(MDODiscipline):
        def __init__(self):
            super().__init__()
            self.input_grammar.update(["z", "y_1"])
            self.output_grammar.update(["y_2"])
            self.default_inputs = {
                "x": ones(1),
                "z": array([4.0, 3.0]),
                "y_1": ones(1),
                "y_2": ones(1),
            }

        def _run(self):
            z, y_1 = self.get_inputs_by_name(["z", "y_1"])
            self.local_data["y_2"] = array([abs(y_1[0]) + z[0] + z[1]])









.. GENERATED FROM PYTHON SOURCE LINES 160-167

Create and execute the scenario
-------------------------------

Instantiate disciplines
^^^^^^^^^^^^^^^^^^^^^^^
We can now instantiate the disciplines
and store the instances in a list which will be used below.

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.. code-block:: default


    disciplines = [Sellar1(), Sellar2(), SellarSystem()]








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Create the design space
^^^^^^^^^^^^^^^^^^^^^^^
In this section,
we define the design space which will be used for the creation of the MDOScenario.
Note that the coupling variables are defined in the design space.
Indeed,
as we are going to select the IDF formulation to solve the MDO scenario,
the coupling variables will be unknowns of the optimization problem
and consequently they have to be included in the design space.
Conversely,
it would not have been necessary to include them
if we aimed to select an MDF formulation.

.. GENERATED FROM PYTHON SOURCE LINES 183-192

.. code-block:: default


    design_space = DesignSpace()
    design_space.add_variable("x", 1, l_b=0.0, u_b=10.0, value=ones(1))
    design_space.add_variable(
        "z", 2, l_b=(-10, 0.0), u_b=(10.0, 10.0), value=array([4.0, 3.0])
    )
    design_space.add_variable("y_1", 1, l_b=-100.0, u_b=100.0, value=ones(1))
    design_space.add_variable("y_2", 1, l_b=-100.0, u_b=100.0, value=ones(1))








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Create the scenario
^^^^^^^^^^^^^^^^^^^
In this section,
we build the MDO scenario which links the disciplines with the formulation,
the design space and the objective function.

.. GENERATED FROM PYTHON SOURCE LINES 198-202

.. code-block:: default

    scenario = create_scenario(
        disciplines, formulation="IDF", objective_name="obj", design_space=design_space
    )








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Add the constraints
^^^^^^^^^^^^^^^^^^^
Then,
we have to set the design constraints

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.. code-block:: default

    scenario.add_constraint("c_1", "ineq")
    scenario.add_constraint("c_2", "ineq")








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As previously mentioned,
we are going to use finite differences to approximate the derivatives
since the disciplines do not provide them.

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.. code-block:: default

    scenario.set_differentiation_method("finite_differences", 1e-6)








.. GENERATED FROM PYTHON SOURCE LINES 217-223

Execute the scenario
^^^^^^^^^^^^^^^^^^^^
Then,
we execute the MDO scenario with the inputs of the MDO scenario as a dictionary.
In this example,
the gradient-based `SLSQP` optimizer is selected, with 10 iterations at maximum:

.. GENERATED FROM PYTHON SOURCE LINES 223-225

.. code-block:: default

    scenario.execute(input_data={"max_iter": 10, "algo": "SLSQP"})





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

        INFO - 14:47:16:  
        INFO - 14:47:16: *** Start MDOScenario execution ***
        INFO - 14:47:16: MDOScenario
        INFO - 14:47:16:    Disciplines: Sellar1 Sellar2 SellarSystem
        INFO - 14:47:16:    MDO formulation: IDF
        INFO - 14:47:16: Optimization problem:
        INFO - 14:47:16:    minimize obj(x, z, y_1, y_2)
        INFO - 14:47:16:    with respect to x, y_1, y_2, z
        INFO - 14:47:16:    subject to constraints:
        INFO - 14:47:16:       c_1(x, z, y_1, y_2) <= 0.0
        INFO - 14:47:16:       c_2(x, z, y_1, y_2) <= 0.0
        INFO - 14:47:16:       y_1: y_1(x, z, y_2): y_1(x, z, y_2) - y_1 == 0.0
        INFO - 14:47:16:       y_2: y_2(z, y_1): y_2(z, y_1) - y_2 == 0.0
        INFO - 14:47:16:    over the design space:
        INFO - 14:47:16:    +------+-------------+-------+-------------+-------+
        INFO - 14:47:16:    | name | lower_bound | value | upper_bound | type  |
        INFO - 14:47:16:    +------+-------------+-------+-------------+-------+
        INFO - 14:47:16:    | x    |      0      |   1   |      10     | float |
        INFO - 14:47:16:    | z[0] |     -10     |   4   |      10     | float |
        INFO - 14:47:16:    | z[1] |      0      |   3   |      10     | float |
        INFO - 14:47:16:    | y_1  |     -100    |   1   |     100     | float |
        INFO - 14:47:16:    | y_2  |     -100    |   1   |     100     | float |
        INFO - 14:47:16:    +------+-------------+-------+-------------+-------+
        INFO - 14:47:16: Solving optimization problem with algorithm SLSQP:
        INFO - 14:47:16: ...   0%|          | 0/10 [00:00<?, ?it]
        INFO - 14:47:16: ...  60%|██████    | 6/10 [00:00<00:00, 119.48 it/sec, obj=3.18]
        INFO - 14:47:16: Optimization result:
        INFO - 14:47:16:    Optimizer info:
        INFO - 14:47:16:       Status: 8
        INFO - 14:47:16:       Message: Positive directional derivative for linesearch
        INFO - 14:47:16:       Number of calls to the objective function by the optimizer: 7
        INFO - 14:47:16:    Solution:
        INFO - 14:47:16:       The solution is feasible.
        INFO - 14:47:16:       Objective: 3.1833939516400456
        INFO - 14:47:16:       Standardized constraints:
        INFO - 14:47:16:          c_1 = 5.764277943853813e-13
        INFO - 14:47:16:          c_2 = -20.244722233074114
        INFO - 14:47:16:          y_1 = [2.06834549e-15]
        INFO - 14:47:16:          y_2 = [1.98063788e-15]
        INFO - 14:47:16:       Design space: 
        INFO - 14:47:16:       +------+-------------+-------------------+-------------+-------+
        INFO - 14:47:16:       | name | lower_bound |       value       | upper_bound | type  |
        INFO - 14:47:16:       +------+-------------+-------------------+-------------+-------+
        INFO - 14:47:16:       | x    |      0      |         0         |      10     | float |
        INFO - 14:47:16:       | z[0] |     -10     | 1.977638883463326 |      10     | float |
        INFO - 14:47:16:       | z[1] |      0      |         0         |      10     | float |
        INFO - 14:47:16:       | y_1  |     -100    | 1.777638883462956 |     100     | float |
        INFO - 14:47:16:       | y_2  |     -100    | 3.755277766925886 |     100     | float |
        INFO - 14:47:16:       +------+-------------+-------------------+-------------+-------+
        INFO - 14:47:16: *** End MDOScenario execution (time: 0:00:00.097537) ***

    {'max_iter': 10, 'algo': 'SLSQP'}



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Post-process the scenario
^^^^^^^^^^^^^^^^^^^^^^^^^
Finally,
we can generate plots of the optimization history:

.. GENERATED FROM PYTHON SOURCE LINES 230-233

.. code-block:: default

    scenario.post_process("OptHistoryView", save=False, show=False)
    # Workaround for HTML rendering, instead of ``show=True``
    plt.show()



.. rst-class:: sphx-glr-horizontal


    *

      .. image-sg:: /examples/mdo/images/sphx_glr_plot_sellar_001.png
         :alt: Evolution of the optimization variables
         :srcset: /examples/mdo/images/sphx_glr_plot_sellar_001.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /examples/mdo/images/sphx_glr_plot_sellar_002.png
         :alt: Evolution of the objective value
         :srcset: /examples/mdo/images/sphx_glr_plot_sellar_002.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /examples/mdo/images/sphx_glr_plot_sellar_003.png
         :alt: Distance to the optimum
         :srcset: /examples/mdo/images/sphx_glr_plot_sellar_003.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /examples/mdo/images/sphx_glr_plot_sellar_004.png
         :alt: Hessian diagonal approximation
         :srcset: /examples/mdo/images/sphx_glr_plot_sellar_004.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /examples/mdo/images/sphx_glr_plot_sellar_005.png
         :alt: Evolution of the inequality constraints
         :srcset: /examples/mdo/images/sphx_glr_plot_sellar_005.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /examples/mdo/images/sphx_glr_plot_sellar_006.png
         :alt: Evolution of the equality constraints
         :srcset: /examples/mdo/images/sphx_glr_plot_sellar_006.png
         :class: sphx-glr-multi-img






.. rst-class:: sphx-glr-timing

   **Total running time of the script:** ( 0 minutes  1.389 seconds)


.. _sphx_glr_download_examples_mdo_plot_sellar.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example


    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_sellar.py <plot_sellar.py>`

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_sellar.ipynb <plot_sellar.ipynb>`


.. only:: html

 .. rst-class:: sphx-glr-signature

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